T[U(N)] duality web: mirror symmetry and gauge/CFT correspondence

In this talk I will review recently obtained correspondence between partition functions of 2d GLSM and certain correlators of Toda CFT. To illustrate this correspondence I will consider particular example: the partition function of 3d T[U(N)] theory on D^2xS^1. I will start by showing how this theory is mirror self-dual. Then I will demonstrate how reduction of mirror pair down to 2d leads to gauge/CFT correspondence. Finally I will consider simplest case of T[U(2)] theory where all details can be worked out precisely.